Preheat the oven to 350 degrees F (175 degrees C). Brush olive oil over spaghetti squash halves; place on a baking sheet.
Bake in the preheated oven until skin is easily pierced with a fork, about 30 minutes.
Start frying sausage in a large skillet when squash has 5 minutes left to bake. Remove squash from the oven and let cool. Add broccoli, mushrooms, onion, garlic, and sage to the skillet with the sausage; cook and stir until onion softens, 5 to 10 minutes.
Scrape insides of squash into spaghetti strands with a fork. Stir into the sausage mixture until well combined. Season with garlic salt, salt, and pepper. Shave Pecorino Romano cheese on top.